-
Notifications
You must be signed in to change notification settings - Fork 0
/
runga_kuta.py
190 lines (156 loc) · 5.25 KB
/
runga_kuta.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
import numpy as np
from matplotlib import pyplot as plt
from scipy.integrate import solve_ivp
# from numba import jit
plt.rcParams["font.size"] = "16"
def make_solver(f, x_range, method, eps):
#@jit
def ode(y_0):
dim = len(y_0)
y = np.zeros((dim, 1))
x = np.array([0.])
y = y_0
x[0] = x_range[0]
h = eps
delta = eps*h/(x_range[1] - x_range[0])
eps_new = eps
i = -1
while x[-1] < x_range[1]:
i = i + 1
while True:
y1 = solve_small_step(f, x[i], y[:, i], h, method)
y2_tmp = solve_small_step(f, x[i], y[:, i], h/2, method)
y2 = solve_small_step(f, x[i]+h/2, y2_tmp, h/2, method)
dist_between_y1_y2 = np.abs(y1 - y2).max()
if dist_between_y1_y2 < delta:
x = np.hstack((x, np.array([x[-1]+h])))
if dist_between_y1_y2 > delta/2:
# print('adjust h->0.9h')
h = h*0.9
elif dist_between_y1_y2 > delta / 4:
# print('h is good')
pass
else:
# print('adjust h->1.1h')
h = h*1.1
eps_new = eps_new - dist_between_y1_y2/2
delta = eps_new*h/(x_range[1]-x[i])
break
else:
# print('h is big! adjust h->0.5h')
h = h / 2
# Joining the two calculation together
# By completing the formula from the course note for h^4 we are joining now
# the two calculations that we did.
y_n = np.zeros((dim, 1))
if method == 'RK3':
y_n[:,0] = 4 * y2 / 3 - y1 / 3
elif method == 'RK4':
y_n[:,0] = 8 * y2 / 7 - y1 / 7
else:
y_n[:,0] = y2
y = np.hstack((y, y_n))
if x[-1] < x_range[1]:
print('Warning, not reach to the end!')
# print()
# return np.vstack((x.reshape((-1,1)), y))
return x, y
return ode
#@jit
def solve_small_step(f, a, f_a, h, method):
if method == 1:
y_new = f_a + h * f(a, f_a)
elif method == 2:
k = h * f(a, f_a)
y_new = f_a + h * f(a + h / 2, f_a + k / 2)
elif method == 'RK3':
k1 = h * f(a, f_a)
k2 = h * f(a + h / 2, f_a + k1 / 2)
k3 = h * f(a + h, f_a + 2 * k2 - k1)
y_new = f_a + (k1 + 4 * k2 + k3) / 6
elif method == 'RK4':
k1 = h * f(a, f_a)
k2 = h * f(a + h / 2, f_a + k1 / 2)
k3 = h * f(a + h / 2, f_a + k2 / 2)
k4 = h * f(a + h, f_a + k3)
y_new = f_a + (k1 + 2 * k2 + 2 * k3 + k4) / 6
return y_new
def check_ode():
# def f(x,y):
# y = y_0 * np.exp(-2*x_ans)
# return -2*y
def f(x,y):
# y = y_0 * np.exp(-x ** 2 / 2)
return -x*y
y_0 = np.array([3.])
x_0 = 0
x_end = 10
eps = 1e-3
x_RK3, y_ans_RK3 = ode(f, (x_0, x_end), y_0, 'RK3', eps)
x_RK4, y_ans_RK4 = ode(f, (x_0, x_end), y_0, 'RK4', eps)
sol = solve_ivp(f, (x_0, x_end), y_0)
# The exact solution
# y = lambda x: y_0 * np.exp(-2 * x)
y = lambda x: y_0* np.exp(-x**2/2)
plt.figure(1)
plt.plot(x_RK4, y(x_RK4), label='analytical results', linewidth=2)
plt.plot(x_RK4, y_ans_RK4[0,:],'o', label='Runga-Kuta order 4', linewidth=1)
plt.legend()
plt.show()
plt.figure(1)
plt.plot(x_RK4, abs(y_ans_RK4[0,:] - y(x_RK4)), label='Absolute error - My', linewidth=2)
# plt.plot(sol.t, abs(sol.y[0,:] - y(sol.t)), label='Absolute error - Scipy', linewidth=2)
plt.yscale('log')
plt.legend()
plt.show()
def check_system_ode():
#@jit
def f(x, y_array):
return np.array([y_array[1],
-y_array[0]])
x0 = 1.
xf = 100.
y0 = np.zeros((2, 1))
y0[:, 0] = np.array([1., 0.])
eps = 1e-3
ode_solver = make_solver(f, (x0, xf), 'RK4', eps)
tic.tic()
x, y = ode_solver(y0)
tic.toc(restart=True)
x, y = ode_solver(y0)
tic.toc()
y_exact = np.array([np.cos(x), -np.sin(x)])
plt.figure()
plt.plot(x, y.T, 'o')
plt.plot(x, y.T)
plt.legend(['$y_1$', '$y_2$', '$\cos$', '$-\sin$'])
plt.show()
plt.figure()
plt.plot(x, np.abs(y.T/y_exact.T - 1))
plt.legend(['Relative Error'])
plt.yscale('log')
plt.show()
def check_steps_calls():
global count
a = 0.1
b = np.array([-5., -1., 1., 4., 7.])
count = 0
def f(x, y):
global count
count += 1
dydx = 1/(a*np.sqrt(np.pi)) * np.sum( np.exp( -(x-b)**2/a**2 ) )
return dydx
y0 = 0
x_range = (-10, 10)
print('%15s %15s %15s %25s %15s' % ('RK order', 'Requsted accuracy', 'No. of steps', 'No. of function calls', 'Relative error'))
for method in ['RK3', 'RK4']:
for eps in [1e-3, 1e-5, 1e-7, 1e-9]:
count = 0
y_final = ode(f, x_range, [y0], method, eps)[1]
print('%15s %15g %15d %25d %15.4g' % (method[-1], eps, len(y_final[0]), count, np.abs(y_final[0,-1]/5-1)))
if __name__ == '__main__':
# Part 1
# check_ode()
from pytictoc import TicToc
tic = TicToc()
check_system_ode()